Field Weights properties not working as expected
Thank you for posting your question very clearly.
1) Decreasing FORCES weights on the smoke sim panel refers to a GLOBAL VALUE. That means the simulation will STOP reacting to the WIND force if you set it to 0 like you´re using now.

2) If YOU DO want to use a wind force, set back WIND FORCE GLOBAL WEIGHT back to 1. Let´s concentrate on the wind force itself. Click on it and go to the properties>dynamics tab (the little bouncing ball).

There, you should manipulate the force of the wind. Please start with a wind force of 100 units. I cannot reference much of your pictures, since you´ve posted only 1 too many times. But I´ll assume your model is big enough to handle 100 unit power wind.

Press playback and your smoke should be moving TOO strong. Decrease by half (on force parameter type 100/2) you´ll get 50. Then you can see it on realtime.
There´s a lot to cover on the SMOKE DOMAIN (the box around your logo) which is the space where the smoke will get simulated.

The direction of the wind is represented by an arrow on the base, and the power factor by circles, that way you´ll have a visual queue of what you´re doing.

To directly playback the smoke simulation once you´re done using wind (or even testing) you´ll need to FIRST SAVE YOUR FILE on a FOLDER (this is important) and then go to smoke simulation DOMAIN (click the box that surrounds your logo) and on the CACHE tab, click FREE AL SIMULATIONS first, then click: BAKE ALL SIMULATIONS.
This will send your smoke simulation to the local folder you saved your project into (see why it was important to save into a folder?)

All right, you should now have a basic understanding of how to work global weight of forces, what forces do, it´s parameters, and basic functions on the smoke domain (the box where the smoke simulates itself).

Thank you for your consideration to vote me as answer.

BFCT_Schiller